I completely agree with Mr. Cook. I think the "trick" to understanding engraving (if there is a trick) is to ask oneself, "If I were a professional engraver, would I be proud of this engraving as my handiwork?" The Georgian and Victorian metal workers and engravers were skilled (often highly-skilled) professionals, likely apprenticed for years, and many were even artistic. For the most part, they were careful about their work, and took pride in it, and received a fair price for it. As such, if you see a tipstaff with very sloppy engraving you may not be looking at original period work.

Hello Great Dane. My sense is that the tipstaff on the left and the tipstaff in the middle, are authentic. They are likely Victorian. They are rather pedestrian, and would not have great value, being the sort of little tipstaff that the working man office holder would keep in a pocket. It's nice to see that the tipstaff on the left has what appears to be the original purple (perhaps originally red, now discolored) velvet in the crown (meant to mimic the Royal Crown's "Cap of Maintenance").
The tipstaff to the right is a bit more involved, and looks to not have the wear pattern I would expect to see on a 150 year old tipstaff. Having said that, it is perhaps just in an excellent state of preservation. I think it's likely to be authentic, but I am just a tad bit less certain of this one. It does, however, look like it's perhaps missing its finial top (the "Maltese Cross"). You would have to send a photo that shows the top clearly for us to be able to determine that with certainty. I think the tipstaff on the right is probably also right, and would have had a slightly higher value than its brethren to the left, were it not for the missing finial. If the finial is not missing, then perhaps it would have a slightly higher value.
I hope this is of some help to you, and please feel free to ask additional questions if you have them.

Adding to an old post, Gorringe's has in its next sale what they have described as: "An early 19th century brass tipstaff, inscribed Police Office, Bow.Stn 2, with turned hardwood handle, 7.5in. Estimate £350-450"
I suppose one has to make up one's own mind as to the authenticity of this item....
Mercifully, Gorringe's provided many photos to peruse.
This tipstaff can be seen here: http://auction.gorringes.co.uk/auction-lot-detail/LSEPT17/5

You're asking a very good question, for which I'm not sure there is a good answer. I believe Mervyn Mitton's book has a photo (black and white?) of a small tipstaff that reputedly belonged to a Bow Street Runner. That little tipstaff in Mervyn's book was unremarkable in its features. I am not sure whether the provenance for that little tipstaff was strong, or just hearsay and there is no real way to tell.
There is another Bow Street attributed tipstaff on Alan Cook's website. I don't know if it is in Mr. Cook's collection, or in another collection. That tipstaff looks like a typical Public Office tipstaff of the late 18th / early 19th Century, and of the type you would want to look at. Mr. Cook says about the tipstaff:
The tipstaff shown in figure 1 is unusual in that it is attributable to a specific officer. It is brass with a turned wood handle and is engraved on the barrel ‘J. Birchall / P.O / Bow Street’. It probably belonged to a John Birchall who appears regularly in the records at the Old Bailey. All of his cases occurred during the reign of George IV between 1823 and 1828. In each he is referred to as either an ‘officer’ or ‘constable’. The type of cases he was involved with included embezzlement, theft, pickpocketing, bigamy and grand larceny.
You can see that tipstaff here: http://www.truncheon.org.uk/middlesex/
There were only half a dozen Bow Street Runners originally in that Public Office, so your chances of finding their tipstaff are slim. They were not particularly wealthy men (although with time they became well-known to the public and the media) and so the likelihood is high that they owned merely a simple, un-engraved, fairly unremarkable tipstaff, which is floating around somewhere, unrecognized for the rarity that it truly is.
There were substantially more Bow Street Horse Patrol mounted riders (about 60 or so to start) and they carried painted truncheons in the period your seek (and up to 1836 when the Horse Patrol amalgamated with the newly-created Metropolitan Police). You may get lucky one day to lay your hands on a painted truncheon from the Bow Street Horse Patrol, but I daresay there are, in my opinion, fewer than ten such truncheons extant.
And so, if you see a "Bow Street" tipstaff offered for sale anywhere, my advice would be to be be very very weary of it as there isn't a very good way to discern reality from fantasy. I believe the chanced of finding a tipstaff with a bona fide iron-clad attribution to a Bow Street Runner is virtually nill.
Having said that, I did see a brass tipstaff engraved "Public Office Bow Street" come to the auction block recently but I would not bid on it because I thought it was spurious. It did not make very much money and so it appears other collectors agreed with me.
The two websites you mention are fairly reputable dealers in militaria who have had both truncheons and tipstaffs for sale, but I don't recall them ever selling a tipstaff attributed to Bow Street.
I'm not sure how helpful my ramblings are, but perhaps you will find them useful in some fashion.

Hats off to all the police men and woman and first responders who stopped the carnage in London a few days ago. Read this article for a great example of the bravery exhibited by all:
http://www.telegraph.co.uk/news/2017/06/05/hero-rookie-officer-took-terrorists-baton-stabbed-face/

Mervyn once sent me photos of a tipstaff he was selling, and I agreed to buy it. He then emailed me again saying he thinks I should also buy another one he was selling, but I demurred. But Mervyn persisted, saying he thought I should really, really, really buy the second one as well because it was too good to be left behind. He said he was so sure about this, that if I didn't like the second tipstaff when I received it, that he would not only take it back, but that he would pay for shipping both ways. I relented, buying both. When I received the parcel I was delighted to see that Mervyn was absolutely right, and this second was not only a great tipstaff, but one of the best, ever. I was delighted he “forced” me to buy it!

Dublin's coat of arms is also three castles. And Wilson is a fairly common Irish surname. Could be an Irish tipstaff. Would make this a very rare piece indeed. Only thing is, the castles don't look exactly right for Dublin.